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I start with an inequality such as:

$$(x-1)(x+4) \geq 0$$

My understanding is that from this point it is solved using the null factor law where: \begin{align*} \begin{cases} x - 1 \geq 0\\\\ x + 4 \geq 0 \end{cases} \end{align*}

This gives me $x \geq 1$ or $x \geq -4$, but the apparent answer to this problem should be $x \geq 1$ or $x \leq -4$.

I don't understand why the inequality sign is flipping in the 2nd term as as far as I know we are not dividing or multiplying by a negative here.

Ronin
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  • @DavidK Yes you're right, I meant to write "or" I've since updated the post. – Ronin Mar 17 '22 at 03:45
  • Now we need to parse "$x \geq 1$ or $x \geq -4$." Usually I would expect this to mean that any $x$ with $x\geq 1$ is in the solution set, and also any $x$ with $x \geq -4$ is in the solution set. Does that work? For example, $0$ is a number that is greater than or equal to $-4$; is the original inequality true if $x = 0$? – David K Mar 17 '22 at 03:50
  • The "null factor law" (or "zero product property", as I know it) applies to when we have $pq = 0$ (not $pq \ge 0$), and it implies $p = 0$ or $q = 0$. The idea is that, if $p \neq 0$, then we can divide both sides by $p$, which gives us $q = 0$. The same proof doesn't work for $pq \ge 0$: if $p < 0$, then it doesn't imply that $q \ge 0$, but instead $q \le 0$ (as dividing by a negative reverses the sign of the inequality). – Theo Bendit Mar 17 '22 at 07:12

1 Answers1

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HINT

When the product of two real numbers is non-negative?

Answer: when both are non-negative or when both are non-positive.

You have considered the first case, but it remains to consider the second situation.

Can you apply such principle to the proposed exercise?

Átila Correia
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